A Comparison of Univariate Smoothing Models: Application to Heart Rate Data Marcus Beal, Member, IEEE

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1 A Comparso of Uvarate Smoothg Models: Applcato to Heart Rate Data Marcus Beal, Member, IEEE E-mal: Abstract There are a umber of uvarate smoothg models that ca be appled to a varety of olear data sets. It s ot always clear whch model wll gve results that are desrable to a gve data set. The focus of ths project s to exame fve uvarate smoothg models to see how they compare to a cotrolled test source. The test source ths case s a model that smulates both the stataeous heart rate (IHR) ad the terbeat terval (IBI) of a perso wth ormal sus rhythm. The smoothg models that are compared are the followg: polyomal, cubc smoothg sple, erel, weghted average, ad local lear. The report wll show that the polyomal smoother s a adequate smoothg model for ths applcato, ad the other four all gve very good results, but the cubc smoothg sple gves the best results for ths applcato. Idex Terms Istataeous heart rate, terbeat terval, predcto error, average squared error I. Itroducto Spectral aalyss of heart rate sgals has bee used a umber of studes, ad obtag accurate models to terpret the ouform sgals s mportat. It s commo for aalysts to use terpolato techques to produce estmates for the stataeous heart rate (IHR) [], []. However, t was show by Lagua et al. that the effects of these terpolato methods caused a atteuato of hgher frequecy compoets []. Recet wor doe volvg artfact removal spectral aalyss of a IHR seres usg a mpulse rejecto flter [3], apples a uvarate smoothg model as part of the process. The smoothg model used ths process was a Kerel smoother usg a Gaussa Kerel. The questos posed ths project were s there a optmal smoother for ths applcato? ad what are the optmal smoothg parameters for the models? The subject of ths project s specfc to the data beg aalyzed because ay gve model wll behave dfferetly depedg o how the data s dstrbuted. For ths reaso there s lttle formato about comparg the behavor of the dfferet smoothg models that s specfc to heart rate. It s typcally ot practcal to perform multple tests smultaeously wth dfferet smoothg models, ad performg a test usg oe specfc model followed by a separate test usg a dfferet model wth the same set of data could geerate results that are based. The focus of ths project s to compare how fve dfferet smoothg models wll behave o a gve set of data from a terbeat terval (IBI) seres. The real challege of ths project was how the models would be compared. Because IBI s a seres of dscretely sampled pots from the true heart rate, there s o formato about the sgal betwee the terbeats. For ths reaso the heart rate was smulated as a model. Ths wor was completed as part of a course project for Egeerg Data Aalyss ad Modelg at Portlad State Uversty durg fall term of 4. A. Data Geerato II. Methodology The heart rate model used for ths project was a seres of four vectors, two that modeled the behavor of a heart rate ad two that were a sampled verso of the heart rate. Both sets of vectors were created as a fucto Matlab by Dr. James McNames, where the put was used as a tme scale ad the output represets a frequecy, as llustrated by Fg. Essetally the scatter plot represets the IBI, whch wll be the data used for creatg the smoothg models, ad the cotuous plot represets the stadard for whch each smoothg model wll be compared to tme(s) heart rate sample Fg s a example of data geerato for the model producg IHR ad IBI data. The sampled data s approxmately /5 the umber of data pots that represet the cotuous fucto. B. Data Modelg Smoothg s a techque that s used to geerate a model for a gve data set wth a sgle put varable x. The geeral dea s to fd the best fucto g( that mmzes the predcto error (PE) o ew puts. For ths project the fve smoothg models cosdered are, polyomal, cubc, erel, weghted averagg ad lear local. a. Polyomal Smoothg Polyomal smoothg s ofte a poor choce for creatg a model as wll be llustrated later o, but t was decded to geerate results based o data rather tha mag pror assumptos. The polyomal smoother taes the form of the followg, p g( ω x

2 where ths fucto s ft to the data usg lear modelg methods. Ths s cosstet because the parameters ω ca be lear whle havg olear puts. Ths ca also be thought of as a lear model wth p dfferet puts where the th put s gve by x x. The p- parameter determes the order of the polyomal, whch s user specfed. b. Cubc Smoothg Sple Cubc sples geeral are of the form, g ( x ) ω 3 x where ule the polyomal smoother the ω parameters are also a fucto of x. Cubc sples also exhbt the followg propertes: g(x ) s cotuous The frst dervatve s cotuous The secod dervatve s cotuous g(x ) y (.e. terpolato of data) Smoothg sples o the other had do ot requre g(x ) y, but rather g(x ) y. Wth ths costrat relaxed there ow must be cosderato to the trade off betwee bas ad varace. Oe way to do ths s to fd the g(x ) that mmzes the followg crtero: E λ ( y g( x )) + λ + '' ( g( ) Ths cotrasts the cubc sple the sese that the frst term was requred to be zero. The secod term represets a roughess pealty ad λ s a user-specfed parameter that cotrols the basvarace tradeoff. Matlab has a bult- fucto to create a cubc smoothg sple where α replaces λ ad rages from zero (lear least squares ft) to oe (cubc sple terpolato). The exact relatoshp betwee α ad λ s ot mportat, but what s, s the uderstadg that α s versely related to λ ad scaled such that t rages from zero to oe. c. Kerel Smoothg The Kerel smoother stems from the defto of Bayes rule whch states: A B] B] P [ B A] A] It s also ow that the g(x ) that mmzes the MSE s gve by: E Y [ Y X x] By expressg Bayes rule the form of a cotuous represetato, a estmate of the above term ca be determed. The process to estmate the above expectato volves a summg of dvdual erels that have a specfed wdth deoted σ, such that b σ (u) represets the shape of the erel. There are a umber of dfferet shapes to represet a erel, but the oe chose here s the same as the oe used the procedure for artfact removal metoed before, whch was the Gaussa erel gve by: u bσ ( u ) ce The fucto for the erel smoother s gve by: dx E[ y x] y b ( x x ) b ( x x ) For the erel smoother, the parameter σ s the oe that affects the smoothess of the model. As σ approaches the g( y ave, but whe σ equals zero the model does a earest eghbor terpolato (at least for erels that have fte support, such as the Gaussa erel). d. Weghted Averagg I geeral local averagg s smlar to the erel smoother. I fact the erel smoother ca be vewed as a weghted average. g ( σ σ ω ( y The real dfferece here s that local averagg uses the -earest eghbors of x to represet the model such that: g( y c ( ) where c() s the data set dex of the th earest pot. However, oe of the dsadvatages of the local average s that t s dscotuous. By tweag the local average a cotuous fucto ca be produced the form ow as weghted local average: g b y b c( ) I ths form the represetato for the erel shapes ca be used here as well, but oly the oes that have fte support. Ths model s very smlar to the erel smoother, wth the ma dfferece beg that the wdth of the erel s determed by the -earest eghbors. It s also smlar to the erel smoother whe the smoothg parameter s adjusted. As approaches the sample sze g( y ave ad whe s small t aga does a earest eghbor terpolato. For ths applcato, the bweght fucto was used to represet the erel such that: d b, where d x x c( ) d + s the dstace betwee the put ad the th earest eghbor. e. Local Lear Models It ca be show that for fxed weghted fuctos, ω(, both erel smoothers ad weghted local averagg models mmze the weghted average squared error (ASE). It s true that lear models mmze the ASE, but ths ca also be exteded to clude the weghted ASE. By combg the erel methods ad the local averagg models, the localzed lear models ca be created. Ths s doe by specfyg the weghts so that pots ear the put have the most fluece o the model output. The result of ths model s a lear terpretato that also uses -earest eghbors to

3 3 determe the wdth of the erel. Ths model also dffers from the prevous two that as approach the sample sze t produces the least squares lear soluto, ad whe becomes small t does pecewse lear terpolato. Aga the bweght erel was used. C. Model Comparso All fve smoothg models requre a smoothess parameter that must be determed by some process. The smoothess parameter s selected such that the PE s mmzed. For smplcty all fve smoothg parameters were tested by a tral ad error type process to determe a reasoable rage of values to test. There are aalytcal techques avalable for determg these smoothg parameters, but for ths project they would be too volved ad sce the true sgal s avalable, fdg the predcto error s trval. Oe method of comparso s to vsually compare the smoothg models dvdually agast the IHR model, whch represets the true sgal ths case. Ths method s useful to a pot, but perhaps a more useful method s to graphcally compare the PE as a fucto of the smoothg parameter. There are two thgs that ca be determed from ths. The frst s the value of the smoothg parameter that mmzes the predcto for each model. The secod thg s the smoothg model that produces the best results terms of mmzg the predcto error. All fve models were eyed to Matlab as fuctos ad aalyzed usg the data geeratg model created by Dr. James McNames. The data model was used to geerate depedet sets of data. Ths data was the set through a loop for a gve smoothg model to determe the PE, whch was smply the absolute dfferece betwee the true model ad the smoothg model. Each of the smoothg models was geerated wth dfferet values for the smoothg parameter. The mea value alog wth a rage of values that dcated the 5th to 95th percetle was determed for each of the smoothg models. Sce may of the dfferet smoothg models use dfferet smoothg parameters, they eeded to be represeted o separate graphs. Fg shows a plot of the PE vs. the smoothg parameter for the polyomal smoother. The ceter le shows the mea PE ad the shaded rego represets the 5th to 95th percetle. The secod smoothg model aalyzed was the cubc smoothg sple. Fg 3 shows a plot of the mea PE for the cubc smoothg sple at.474, wth the 5th to 95th percetle ragg from.44 to.57. All of these values correspod to the smoothg parameter that gves the mmum PE. Predcto Error Smoothg Parameter Fg 3 shows a plot of the PE vs. the smoothg parameter for the cubc smoothg sple. The ceter le shows the mea PE ad the shaded rego represets the 5th to 95th percetle. The thrd smoothg model aalyzed was the erel smoother. Fg 4 shows the mea PE for the erel smoother at.536, wth the 5th to 95th percetle ragg from.478 to.57. All of these values correspod to the smoothg parameter that gves the mmum PE. III. Results The frst smoothg model aalyzed was the polyomal smoother. Fg shows a plot of the mea PE for the polyomal smoother at.57 wth the 5th ad 95th percetle ragg from.85 to.8. All of these values correspod to the smoothg parameter that gves the mmum PE..55 Predcto Error Predcto Error Smoothg Parameter Smoothg Parameter Fg 4 shows a plot of the PE vs. the smoothg parameter for the erel smoother. The ceter le shows the mea PE ad the shaded rego represets the 5th to 95th percetle.

4 4 The ext smoothg model aalyzed was the weghted average. Fg 5 shows a plot of the mea PE for the weghted average at.54 wth the 5th to 95th percetle ragg from.498 to.59. All of these values correspod to the smoothg parameter that gves the mmum PE...8 oe to four. I ths partcular demostrato the lowest value for the mea PE s represeted by a fourth order polyomal, Fg 7. As see the plot, the predcto s ot at all close to the true sgal ad should ot be used for ths applcato..8.6 Predcto Error Istaeous Heart Rate (Hz) Smoothg Parameter Fg 5 shows a plot of the PE vs. the smoothg parameter for the weghted average smoother. The ceter le shows the mea PE ad the shaded rego represets the 5th to 95th percetle. The last smoothg model aalyzed was the local lear model. Fg 6 shows a plot of the mea PE for the local lear model at.56 wth the 5th to 95th ragg from.458 to.586. All of these values correspod to the smoothg parameter that gves the mmum PE. Predcto Error Smoothg Parameter Fg 6 shows a plot of the PE vs. the smoothg parameter for the local lear smoother. The ceter le shows the mea PE ad the shaded rego represets the 5th to 95th percetle. IV. Dscusso As expected the polyomal smoother dd a extremely poor job of predctg the true model. All the values for the PE were very close to each other wth the smoothg parameter the rage of Tme (s) Fg 7 shows a fourth order polyomal smoothg model blac agast the true model gray. The other four smoothg models all dd a very good job of predctg the true model. I fact they dd so good that graphcally t s dffcult to dstgush oe sgal from aother whe compared to the true model (Fg 8). From the prevous plots of PE t may appear the there s a larger rage of values the 5th to 95th percetle usg the cubc smoothg sple ad erel smoother, but ths s oly because the rage of values s so much smaller tha that of the polyomal, weghted average ad local lear smoothers. The reaso ths was doe was because the smoothg parameters for the cubc smoothg sple ad erel smoother are a cotuous rage of values, where as the other models are dscrete teger values. Ths meat that the rage of values eeded to be small order to fd the optmal smoothg parameter for the cubc smoothg sple ad erel smoother. The smoothg parameters that mmzed these models were all ear the pot of terpolato. For the cubc smoothg sple ths value correspoded to α.995. As metoed the methodology secto, terpolato for the cubc smoothg sple occurs at oe. For the erel smoother the value of the smoothg parameter σ that mmzed the PE correspoded to σ.34, whch s close to zero, the value correspodg to earest eghbor terpolato. The last two smoothers, weghted average ad local lear, both use a smoothg parameter based o the umber of earest eghbors, ad both requred that ths value correspod to, whch s aga close to the terpolato value of. It should also be oted here that the reaso the plot of the weghted average oly goes as low as s due to a gltch the smulato. Essetally, the bweght fucto metoed the methodology secto would perodcally result a value of zero whe. The problem wth ths s that the estmate would the be udefed at ths value.

5 5 (a) Cubc Smoothg Sple (b) Kerel Smoother tme(s) tme(s) (c) Weghted Local Average (d) Local Lear Average tme(s) tme(s) Fg 8 shows the cubc smoothg sple, the erel smoother, the weghted average, ad the local lear model (a-d) respectvely blue, plotted agast the true model red. There are a umber of useful observatos wth Fg 8. The cubc smoothg sple appears to predct the true model qute well; fact t s more accurate tha ay of the other four models. The erel smoother also does well, but maybe ot as well as the cubc smoothg sple. Ths could be due part to ts more coservatve ature. The erel smoother does ot exceed ay pot greater or less tha the maxmum or mmum value the scatter plot. The weghted average, as expected, s very smlar to the erel smoother ad t to has a smlar coservatve ature. The last s the local lear average, whch also does a reasoable job of predctg the true model ad maybe a slghtly better job tha the erel or the weghted average. Ths model s ule the two prevous oes because t does ot have ths same coservatveess, although there s a tedecy to overshoot the data wth sharp spes that are udesrable. Upo comparg the PE values t s also evdet that the erel, weghted average or local lear models dd ot stad out as beg sgfcatly better tha each other. However, the cubc smoothg sple dd perform better tha all the other models by a slght marg. Upo rug these smulatos multple tmes t was observed that the mea PE stayed very cosstet, but the 5th to 95th percetle rage fluctuated slghtly. Based o these results the cubc smoothg sple produces a model of a IHR wth the smallest PE. V. Cocluso Ths report shows that terpolato s ot the best applcato for creatg a model for a stataeous heart rate. Although t was show that the cubc smoothg sple, erel, weghted average ad the local lear all geerated very good models, t was the cubc smoothg sple that stood out. Aother advatage for usg the cubc smoothg sple s that Matlab already has ths fucto bult-, so there would be less code for ayoe usg Matlab for ths applcato. Ths report also shows that the values for the smoothg parameters were all very close to terpolato. Whe usg these smoothers as models for a stataeous heart rate they should be set to for the weghted average ad the local lear model, α.995 for the cubc smoothg sple ad σ.34 for the erel smoother. The last thg ths report cocludes s that the polyomal smoother should ever be used as a model for a stataeous heart rate. Acowledgemets The author would to tha Dr. James McNames for hs ad creatg the data geeratg model for ths report, ad for hs advce pursg the results for ths project. Further recogto should also go to Dr. McNames for supplyg the formulas ad demostratg how to apply them hs lecture otes at Portlad State Uversty. Refereces [] Tas Force of the Europea Socety of Cardology ad the North Amerca Socety of Pacg ad Electrophysology. Heart Rate varablty Stadards of Measuremet, Physologcal Iterpretato, ad Clcal Use. Crculato 996, p [] P. Lagua, G.B. Moody, R.G. Mar. Power Spectral Desty of Uevely Sampled Data by Least-square Aalyss: Performace ad Applcato to Heart Rate Sgals. IEEE Tras Bomed Eg, Vol. 45, o.6, pp , Jue 998. [3] J. McNames, T. Thog, M. Aboy. Impulse Rejecto Flter for Artfact Removal Spectral Aalyss of Bomedcal Sgals. Preceedgs of the 6th Aual Iteratoal Coferece of the IEE EMBS. Sa Fracsco, CA, USA. Pp September -5 4.